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Does a monotone subadditive $f: \mathcal{P}(\bf N)\to [0,1]$ admit a finite partition with values in $(0,1)$?
Right, I overlooked something in my thoughtless use of the Cauchy test, and then removed my previous comment. 
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Does a monotone subadditive $f: \mathcal{P}(\bf N)\to [0,1]$ admit a finite partition with values in $(0,1)$?
@SeanEberhard. The fnc in your comment is sometimes called the upper exponential density; see, e.g., G. G. Lorentz, On a problem of additive number theory, Proc. Amer. Math. Soc. 5 (1954), No. 5, 838841, or Section 4 in: G. Grekos, On various definitions of density (survey), Tatra Mt. Math. Publ. 31 (2005), 1727. But I don't know if it's the same as the fnc in Ilya Bogdanov's example. 
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Does a monotone subadditive $f: \mathcal{P}(\bf N)\to [0,1]$ admit a finite partition with values in $(0,1)$?
Very nice. Incidentally, the fnc in this example is also shiftinvariant, i.e. $f(X + h) = f(X)$ for all $X \subseteq \mathbf N$ and $h \in \mathbf N$ (immediate by the limit comparison test), and has the strong Darboux property, too: If $A\subseteq B\subseteq \mathbf N$ and $\gamma \in {]f(A), f(B)[}$, then $f$ being weakly Darboux implies the existence of a set $C\subseteq B$ s.t. $f(C) = \gamma$, and hence $f(A \cup C) = \max(f(A), f(C)) = \gamma$. Since, of course, $A \subseteq A \cup C \subseteq B$, that is enough to conclude. 
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Does a monotone subadditive $f: \mathcal{P}(\bf N)\to [0,1]$ admit a finite partition with values in $(0,1)$?
@JochenWengenroth. Of course. But only if you can prove that $0 < f((A_1 \cup \cdots \cup A_k)^c) < 1$ for some $k$, in your construction. In principle, you don't have any information on the behaviour of $f(A^c)$ for $A \subseteq \mathbf N$ and $f(A) > 0$ (if $f(A) = 0$, it is seen that $f(A^c) = 1$). 
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Sep
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Reference request: Darboux properties of realvalued set functions (measures, densities, etc.)
@Nate Eldredge. The question is a little bit unclear, indeed. I'm basically interested in understanding if (ii) has been called under a different name in the literature (either on measures or on densities). To put it in other terms, Q1 is more about keywords I can use to track the literature on the subject, whereas Q2 is more about actual results related to properties (i) and (ii). And again, I'm mostly interested in the case of densities, for which I couldn't really find that much. 
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Reference request: Darboux properties of realvalued set functions (measures, densities, etc.)
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Sep
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revised 
Reference request: Darboux properties of realvalued set functions (measures, densities, etc.)
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Sep
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Reference request: Darboux properties of realvalued set functions (measures, densities, etc.)
@Dave L Renfro. Thanks. From what I can say, [B] is essentially focused on functions from $\mathbf R^n$ to (the ground set of) a metric space $\mathcal X$, for which the authors introduce, and study, a kind of Darboux property that is ultimately shaped by the Euclidean structure of the domain and the metric structure of $\mathcal X$. Not really what I'm looking for. (I don't have access to [A], right now.) 
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Reference request: Darboux properties of realvalued set functions (measures, densities, etc.)
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Sep
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Reference request: Darboux properties of realvalued set functions (measures, densities, etc.)
I don't know! (I know nothing about continuum theory.) But thanks for the pointer, I'll give it a try. 
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Reference request: Darboux properties of realvalued set functions (measures, densities, etc.)
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Reference request: Darboux properties of realvalued set functions (measures, densities, etc.)
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Sep
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asked  Reference request: Darboux properties of realvalued set functions (measures, densities, etc.) 
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Sep
22 
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Approximating integers with prime quotients
Yes, that's it! 
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Approximating integers with prime quotients
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Approximating integers with prime quotients
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Sep
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answered  Approximating integers with prime quotients 